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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2410.11246 |
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| _version_ | 1866909349926928384 |
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| author | Tang, Quanyu Wang, Wei Zhang, Hao |
| author_facet | Tang, Quanyu Wang, Wei Zhang, Hao |
| contents | A \emph{rational orthogonal matrix} $Q$ is an orthogonal matrix with rational entries, and $Q$ is called \emph{regular} if each of its row sum equals one, i.e., $Qe = e$ where $e$ is the all-one vector. This paper presents a method for generating all regular rational orthogonal matrices using the classic Cayley transformation. Specifically, we demonstrate that for any regular rational orthogonal matrix $Q$, there exists a permutation matrix $P$ such that $QP$ does not possess an eigenvalue of $-1$. Consequently, $Q$ can be expressed in the form $Q = (I_n + S)^{-1}(I_n - S)P$, where $I_n$ is the identity matrix of order $n$, $S$ is a rational skew-symmetric matrix satisfying $Se = 0$, and $P$ is a permutation matrix. Central to our approach is a pivotal intermediate result, which holds independent interest: given a square matrix $M$, then $MP$ has $-1$ as an eigenvalue for every permutation matrix $P$ if and only if either every row sum of $M$ is $-1$ or every column sum of $M$ is $-1$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_11246 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | The Generation of All Regular Rational Orthogonal Matrices Tang, Quanyu Wang, Wei Zhang, Hao Combinatorics 05C50 A \emph{rational orthogonal matrix} $Q$ is an orthogonal matrix with rational entries, and $Q$ is called \emph{regular} if each of its row sum equals one, i.e., $Qe = e$ where $e$ is the all-one vector. This paper presents a method for generating all regular rational orthogonal matrices using the classic Cayley transformation. Specifically, we demonstrate that for any regular rational orthogonal matrix $Q$, there exists a permutation matrix $P$ such that $QP$ does not possess an eigenvalue of $-1$. Consequently, $Q$ can be expressed in the form $Q = (I_n + S)^{-1}(I_n - S)P$, where $I_n$ is the identity matrix of order $n$, $S$ is a rational skew-symmetric matrix satisfying $Se = 0$, and $P$ is a permutation matrix. Central to our approach is a pivotal intermediate result, which holds independent interest: given a square matrix $M$, then $MP$ has $-1$ as an eigenvalue for every permutation matrix $P$ if and only if either every row sum of $M$ is $-1$ or every column sum of $M$ is $-1$. |
| title | The Generation of All Regular Rational Orthogonal Matrices |
| topic | Combinatorics 05C50 |
| url | https://arxiv.org/abs/2410.11246 |